Shortly after arriving at Manchester United this summer in a £28million switch from Southampton, Shaw was ordered away from the main group to work on his physical conditioning.
Van Gaal said at the time that the teenage full-back was ‘not very fit’ and unable to ‘perform’ to the standard expected of him.
Hodgson has now revealed that Shaw was offered similar advice during his time at St Mary’s and again during the World Cup finals in Brazil – where he made just one appearance.
He said: “I must say watching him play in the matches for Southampton, he didn’t seem to have too many problems coping with the physical side of the game.
“But that doesn’t mean to say he can’t be fitter and if Louis thinks that he can get him fitter, then I’m sure he will certainly have our approval.
“I think when he (Van Gaal) talks to (former Southampton boss Mauricio) Pochettino, then Pochettino will say to him, ‘What you were saying to Luke Shaw is what I was actually saying to him’.
“He (Shaw) might have had it mentioned to him a couple of times during the World Cup when we were together by people like myself and (England physio) Gary Lewin.”
Shaw is currently nursing a hamstring injury which has prevented him from making his competitive debut for United and from figuring in England’s upcoming double-header against Norway and Switzerland.
Hodgson believes the youngster will have to work just as hard when he shakes off the knock to ensure that he is ready to compete at the highest level on the domestic and international stages.
He added: “I think a lot of Luke’s situation is purely down to the fact that he is still young, his body is changing, but I think he’s aware that the work rate which is required does mean that he’s got to work very, very hard on that aspect of his game.
“I think he’s been aware of it, Luke, and he will get fitter and stronger, not least because he’s still a very young man and his body has not yet reached its proper proportions for the level he wants to play at.”